This isn't done. If you're seeing this, it's because I linked to it in another writing, in which case what's currently here should serve for the purposes of that writing.

The Roles Played by Intuition

We use the word intuition in many different ways, primarily for those pre- or non-verbal aspects of cognition. I'll be using the word to denote the structural factors of conscious experience, the things which your cognitions are molded by. Here, I'll outline two complex dichotomies of intuition which have appeared to me upon reflection on my own thought processes.

A note: I'm using the word "cognition" here in a way broader than its usual meaning, to mean deliberate mental processes in general. Maybe a better word will come to me later, in which case I'll go back and edit this.

Two Dichotomies of Intuition

First, the abstract/sensory dichotomy.

Abstract intuition concerns "pictures of situations", or forms according to which various facts and their relations are situated. This is the kind of intuition one builds over the course of developing expertise about some abstract topic: you understand "why" a particular (say, mathematical) gadget exists, what the idea behind it is, where it's useful, and so on.
Sensory intuition concerns "models of worlds of sensorimotor activity". Video games which one has played for a long time offer very good examples. This mental model is largely causal and mechanical, just as our view of the physical world insofar as we can interact with it is; you have an idea of what might happen if you do any given thing.

(In some worlds of activity, such as multiplayer games, this causal model is statistical; you intuitively construct a probability density and marginalize your actions over it. This statistical description seems to fail a basic plausibility test — nobody can consciously do this fast enough to effectively use it in e.g. a shooter — but, upon reflection on my own cognitive processes as I play some games I'm experienced in, this is indeed what seems to happen. There are almost certainly some very intricate neural mechanisms operating behind the scenes here).

Second, the perceptive/active dichotomy.

Perceptive intuition concerns the maintenance of an intuitive model, whether abstract or perceptory, and schemata for determining whether cognitions concord with these intuitive models. This kind of intuition is being utilized when you notice that someone's face looks odd but can't quite articulate why, or have a strange feeling about the unusual dimensions of a house, as well as when you find a mathematical concept being utilized in a way your abstract model of it suggests it isn't meant to be utilized. (In particular, it underlies the "tacit knowledge" of experts).
Active intuition concerns schemata for making cognitions concord with intuitive models, and schemata for generating new cognitions using intuitive models, i.e. using intuition to guide thought. When I decide how to word this very sentence, for instance, I use active intuition to turn my ideas into words that conform to the grammatical rules that are suggested by my perceptive-intuitive model of English (which is my first language; I couldn't tell you the formal rules of this language even if I wanted to).

Four Quadrants of Intuition

These two dichotomies are perpendicular to one another, so that we may use them to talk about 2 x 2 = 4 kinds of intuition:

Abstract-perceptive intuition, for instance the ability to sense when something feels off in a scientific paper written by a crank.
Abstract-active intuition, for instance the often non-cognitive ability to determine the right mathematical object to use in order to solve a problem.

A great example, from the Wikipedia page on Srinivasa Ramanujan:

Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied.
Sensory-perceptive intuition, for instance the ability to sense when something is off in some 'sensorimotor world of activity' you have experience in, such as the judgement of faces. (I choose this example because it's something everyone has natural experience in. But if you're an NBA player, you might notice something off when you're made to play with a ball that's just a bit smaller than regulation size — same idea).
Sensory-active intuition, for instance the ability to know how fast you should go in order to jump across a gap, or how to move in order to position an enemy in an inconvenient spot in a video game. (Knowing what spots are inconvenient requires sensory-perceptive intuition).

Furthermore, both the abstract-sensory and the perceptive-active dichotomies are both dichotomies: in general, intuitions will arise through an interplay of these various ideals. They’re complex dichotomies as well, so such an interplay depends not on the specific weights of the combinations of these ideal types but on noncommutative patterns of interaction. What one might call an intuitive grasp of a video game, for instance, is really a sensory action-perception loop, combining both sensory-perceptive and sensory-active intuitions in order to build and act on an internal model of the game in real time. My developing this theory (not just writing about it, but mentally having and working over the ideas) requires an abstract action-perception loop.